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CHAPTER 14: OPTIONS MARKETS
1.
Call option, X = 85
Put option, X = 85
Call option, X = 90
Put option, X = 90
Call option, X = 95
Put option, X = 95
Cost
3.82
0.15
0.40
1.80
0.05
6.30
Payoff
5.00
0.00
0.00
0.00
0.00
5.00
Profit
1.18
-0.15
-0.40
-1.80
-0.05
-1.30
2.
c is false. This is the description of the payoff to a put, not a call.
3.
c is the only correct statement.
4.
Each contract is for 100 shares: $7.25  100 = $725
5.
In terms of dollar returns:
Stock price:
All stocks (100 shares)
All options (1,000 shares)
Bills + 100 options
Price of Stock Six Months From Now
$80
$100
$110
$120
8,000 10,000 11,000 12,000
0
0 10,000 20,000
9,360
9,360
10,360 11,360
In terms of rate of return, based on a $10,000 investment:
Price of Stock Six Months From Now
Stock price:
$80
$100
$110
$120
All stocks (100 shares)
-20%
0%
10%
20%
All options (1,000 shares) -100%
-100%
0%
100%
Bills + 100 options
-6.4%
-6.4%
3.6%
13.6%
14-1
6.
7.
a.
Purchase a straddle, i.e., both a put and a call on the stock. The total cost of
the straddle would be: $10 + $7 = $17
b.
Since the straddle costs $17, this is the amount by which the stock would have
to move in either direction for the profit on either the call or the put to cover
the investment cost (not including time value of money considerations).
a.
Sell a straddle, i.e., sell a call and a put to realize premium income of:
$4 + $7 = $11
b.
If the stock ends up at $50, both of the options will be worthless and your
profit will be $11. This is your maximum possible profit since, at any other
stock price, you will have to pay off on either the call or the put. The stock
price can move by $11 (your initial revenue from writing the two at-themoney options) in either direction before your profits become negative.
c.
Buy the call, sell (write) the put, lend the present value of $50. The payoff
is as follows:
Position
Long call
Short put
Lending
Total
Initial Outlay
C=7
-P = -4
50/(1 + r)(1/4)
7 – 4 + [50/(1 + r)(1/4)]
Final Payoff
ST < X
ST > X
0
ST – 50
-(50 – ST)
0
50
50
ST
ST
The initial outlay equals: (the present value of $50) + $3
In either scenario, you end up with the same payoff as you would if you
bought the stock itself.
14-2
8.
a.
By writing covered call options, Jones receives premium income of €30,000.
If, in January, the price of the stock is less than or equal to €45, he will keep
the stock plus the premium income. Since the stock will be called away from
him if its price exceeds €45 per share, the most he can have is:
€450,000 + €30,000 = €480,000
(We are ignoring interest earned on the premium income from writing the
option over this short time period.) The payoff structure is:
Stock price
Less than €45
Greater than €45
Portfolio value
(10,000 times stock price) + €30,000
€450,000 + €30,000 = €480,000
This strategy offers some premium income but leaves the investor with
substantial downside risk. At the extreme, if the stock price falls to zero,
Jones would be left with only €30,000. This strategy also puts a cap on the
final value at €480,000, but this is more than sufficient to purchase the house.
b.
By buying put options with a €35 strike price, Jones will be paying €30,000 in
premiums in order to insure a minimum level for the final value of his
position. That minimum value is: (€35  10,000) – €30,000 = €320,000
This strategy allows for upside gain, but exposes Jones to the possibility of a
moderate loss equal to the cost of the puts. The payoff structure is:
Stock price
Less than €35
Greater than €35
c.
Portfolio value
€350,000 – €30,000 = €320,000
(10,000 times stock price) – €30,000
The net cost of the collar is zero. The value of the portfolio will be as follows:
Stock price
Less than €35
Between €35 and €45
Greater than €45
Portfolio value
€350,000
10,000 times stock price
€450,000
If the stock price is less than or equal to €35, then the collar preserves the
€350,000 in principal. If the price exceeds €45, then Jones gains up to a cap
of €450,000. In between €35 and €45, his proceeds equal 10,000 times the
stock price.
The best strategy in this case is (c) since it satisfies the two requirements
of preserving the €350,000 in principal while offering a chance of getting
€450,000. Strategy (a) should be ruled out because it leaves Jones
exposed to the risk of substantial loss of principal.
Our ranking is:
(1) c
(2) b
(3) a
14-3
9.
Bearish spread
Position
Long call (X2)
Short call (X1)
S < X1
X1 < S < X2
S > X2
0
0
0
–(S – X1)
S – X2
–(S – X1)
0
X1 – S
X1 – X2
Total
In the bullish spread, the payoff either increases or is unaffected by stock price
increases. In the bearish spread, the payoff either increases or is unaffected by
stock price decreases.
10.
a.
Butterfly Spread
Position
Long call (X1)
Short 2 calls (X2)
Long call (X3)
Total
S < X1
X1 < S < X2
X2 < S < X3
X3 < S
0
0
0
S – X1
S – X1
–2(S – X2)
S – X1
–2(S – X2)
S – X3
0
0
0
S – X1
14-4
0
2X2 – X1 – S
(X2–X1 ) – (X3–X2) = 0
b.
Vertical combination
Position
Long call (X2)
Long put (X1)
S < X1
X1 < S < X2
S > X2
0
X1 – S
0
0
S – X2
Total
X1 – S
0
0
S – X2
11. The executive receives a bonus if the stock price exceeds a certain value, and
receives nothing otherwise. This is the same as the payoff to a call option.
12.
The farmer has the option to sell the crop to the government, for a guaranteed
minimum price, if the market price is too low. If the support price is denoted PS
and the market price PM then we can say that the farmer has a put option to sell the
crop (the asset) at an exercise price of PS even if the market price of the underlying
asset (PM) is less than PS.
13.
[Note: Problem 13(a) in the text should read, “Plot the payoff and profit diagrams
to a straddle position with an exercise (strike) price of $130.” Therefore, in the
Excel spreadsheet and diagrams on the next two pages, the straddle position is
shown for an exercise price of $130, not $115.]
The Excel spreadsheet for both parts (a) and (b) is shown on the next page, and the
profit diagrams are on the following page.
14-5
13. a. & b.
Stock Prices
Beginning Market Price
Ending Market Price
116.5
130
Ending
Buying Options:
Call Options Strike
110
120
130
140
Price
22.80
16.80
13.60
10.30
Payoff
20.00
10.00
0.00
0.00
Profit
-2.80
-6.80
-13.60
-10.30
Return %
-12.28%
-40.48%
-100.00%
-100.00%
Put Options Strike
110
120
130
140
Price
12.60
17.20
23.60
30.50
Payoff
0.00
0.00
0.00
10.00
Profit
-12.60
-17.20
-23.60
-20.50
Return %
-100.00%
-100.00%
-100.00%
-67.21%
Straddle
110
120
130
140
Price
35.40
34.00
37.20
40.80
Payoff
20.00
10.00
0.00
10.00
Profit
-15.40
-24.00
-37.20
-30.80
Return %
-43.50%
-70.59%
-100.00%
-75.49%
110
120
130
140
Price
22.80
16.80
13.60
10.30
Payoff
-20
-10
0
0
Profit
2.80
6.80
13.60
10.30
Return %
12.28%
40.48%
100.00%
100.00%
110
120
130
140
Price
12.60
17.20
23.60
30.50
Payoff
0
0
0
10
Profit
12.60
17.20
23.60
40.50
Return %
100.00%
100.00%
100.00%
132.79%
Price
Payoff
Profit
16.80
13.60
10.00
0
10.00
Stock Price
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
210
Selling Options:
Call Options Strike
Put Options Strike
Money Spread
Bullish Spread
Purchase 120 Call
Sell 130 Call
Combined Profit
14-6
-6.80
13.60
6.80
X 130 Straddle
Profit
Ending
Stock Price
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
210
-37.20
42.80
32.80
22.80
12.80
2.80
-7.20
-17.20
-27.20
-37.20
-27.20
-17.20
-7.20
2.80
12.80
22.80
32.80
42.80
Bullish
Spread
6.80
-3.2
-3.2
-3.2
-3.2
-3.2
-3.2
-3.2
-3.2
6.8
6.8
6.8
6.8
6.8
6.8
6.8
6.8
6.8
Spreads and Straddles
50.00
40.00
30.00
20.00
10.00
130 Straddle
0.00
0
50
100
150
200
250
Bullish Spread
-10.00
10.00
-20.00
20.00
-30.00
30.00
-40.00
40.00
-50.00
50.00
Stock Price
14. The bondholders have, in effect, made a loan which requires repayment of B
dollars, where B is the face value of bonds. If, however, the value of the firm (V) is
less than B, then the loan is satisfied by the bondholders taking over the firm. In
this way, the bondholders are forced to “pay” B (in the sense that the loan is
cancelled) in return for an asset worth only V. It is as though the bondholders wrote
a put on an asset worth V, with exercise price B. Alternatively, one can view the
bondholders as giving to the equity holders the right to reclaim the firm by paying
off the B dollar debt. The bondholders have issued a call to the equity holders.
15. a.
Donie should choose the long strangle strategy. A long strangle option
strategy consists of buying a put and a call with the same expiration date and
the same underlying asset, but different exercise prices. In a strangle strategy,
the call has an exercise price above the stock price and the put has an exercise
price below the stock price. An investor who buys (goes long) a strangle
expects that the price of the underlying asset (TRT Materials in this case) will
either move substantially below the exercise price on the put or above the
exercise price on the call. With respect to TRT, the long strangle investor
buys both the put option and the call option for a total cost of $9.00, and will
experience a profit if the stock price moves more than $9.00 above the call
exercise price or more than $9.00 below the put exercise price. This strategy
would enable Donie's client to profit from a large move in the stock price,
either up or down, in reaction to the expected court decision.
14-7
b.
i. The maximum possible loss per share is $9.00, which is the total cost of the
two options ($5.00 + $4.00).
ii. The maximum possible gain is unlimited if the stock price moves outside
the breakeven range of prices.
iii. The breakeven prices are $46.00 and $69.00. The put will just cover costs
if the stock price finishes $9.00 below the put exercise price ($55 − $9 = $46),
and the call will just cover costs if the stock price finishes $9.00 above the call
exercise price ($60 + $9 = $69).
16. a.
ST < 1040
ST > 1040
ST
ST
1040 – ST
1040
0
ST
ST < 1120
ST > 1120
Bills
Call
1120
0
1120
ST – 1120
Total
1120
ST
Protective Put
Stock
Put
Total
Bills and Call
Payoff
Bills plus calls
(Dashed line)
1120
1040
Protective put strategy
(Solid line)
ST
1040
1120
14-8
b.
The bills plus call strategy has a greater payoff for some values of ST and
never a lower payoff. Since its payoffs are always at least as attractive and
sometimes greater, it must be more costly to purchase.
c.
The initial cost of the stock plus put position is $1,208 and the cost of the
bills plus call position is $1,240.
Position
ST = 0
Stock
+ Put
0
1040
Payoff
Profit
ST = 1040
ST = 1120
ST = 1200
ST = 1280
1040
0
1120
0
1200
0
1280
0
Position
1040
-168
ST = 0
1040
-168
ST = 1040
1120
-88
ST = 1120
1200
-8
ST = 1200
1280
+72
ST = 1280
Bill
+ Call
1120
0
1120
0
1120
0
1120
80
1120
160
Payoff
Profit
1120
-120
1120
-120
1120
-120
1200
-40
1280
+40
Profit
Protective put
Bills plus calls
1040
1120
ST
-120
-168
d.
The stock and put strategy is riskier. It does worse when the market is down,
and better when the market is up. Therefore, its beta is higher.
14-9
17.
a.
Joe’s strategy
Position
Stock index
Long put (X = 1200)
Total
Profit = payoff – 1260
Initial Outlay
1200
60
1260
Final Payoff
ST < 1200 ST > 1200
ST
ST
1200 – ST
0
1200
ST
-60
ST – 1260
Initial Outlay
1200
45
1260
Final Payoff
ST < 1170 ST > 1170
ST
ST
1170 – ST
0
1170
ST
-75
ST – 1245
Sally’s Strategy
Position
Stock index
Long put (X = 1170)
Total
Profit = payoff – 1245
Profit
Sally
1170
1200
Joe
ST
-60
-75
18.
b.
Sally does better when the stock price is high, but worse when the stock
price is low. (The break-even point occurs at S = $1185, when both
positions provide losses of $60.)
c.
Sally’s strategy has greater systematic risk. Profits are more sensitive to the
value of the stock index.
a.
If an investor buys a call option and writes a put option on a T-bond, then, at
maturity, the total payoff to the position is (ST – X), where ST is the price of the
T-bond at the maturity date (time T) and X is the exercise price of the options.
This is equivalent to the profit on a forward or futures position with futures
price X. If you choose an exercise price (X) equal to the current T-bond futures
price, then the profit on the portfolio replicates that of market-traded futures.
14-10
b.
Such a position would increase the portfolio duration, just as adding a T-bond
futures contract increases duration. As interest rates fall, the portfolio increases
in value, so that duration is longer than it was before the synthetic futures
position was established.
c.
Futures can be bought and sold very cheaply and quickly. They give the
manager flexibility to pursue strategies or particular bonds that seem
attractively priced without worrying about the impact of these actions on
portfolio duration. The futures can be used to make adjustments to duration
necessitated by other portfolio actions.
19. a.
Position
S < 85
85 < S < 90
S > 90
Short call
Short put
0
– (85 –S)
0
0
– (S – 90)
0
S – 85
0
90 – S
Total
Payoff
85
90
Write put
b.
ST
Write call
Proceeds from writing options (from Figure 14.1):
Call = $1.55
Put
= $0.90
Total = $2.45
If IBM is selling at $87, both options expire out of the money, and profit
equals $2.45. If IBM is selling at $95, the call written results in a cash
outflow of $5 at maturity, and an overall profit of: $2.45 – $5.00 = $2.55
14-11
c.
You break even when either the short position in the put or the short position
in the call results in a cash outflow of $2.45. For the put, this requires that:
$2.45 = $85 – S  S = $82.55
For the call this requires that:
$2.45 = S – $90  S = $92.45
d.
20.
The investor is betting that the IBM stock price will have low volatility. This
position is similar to a straddle.
The put with the higher exercise price must cost more. Therefore, the net outlay to
establish the portfolio is positive.
14-12
21.
Buy the X = 62 put (which should cost more than it does) and write the X = 60 put.
Since the options have the same price, the net outlay is zero. Your proceeds at
maturity may be positive, but cannot be negative.
Position
Long put (X = 62)
Short put (X = 60)
ST < 60
60 < ST < 62
ST > 62
62 – ST
– (60 – ST)
62 – ST
0
0
Total
22.
2
0
The following payoff table shows that the portfolio is riskless with time-T value
equal to £10. Therefore, the risk-free rate is: (£10/£9.50) – 1 = 0.0526 = 5.26%
Position
Buy stock
Short call
Long put
Total
23.
0
62 – ST
a.
ST < 10
ST > 10
ST
ST
– (ST – 10)
0
10 – ST
10
0
10
Conversion value of a convertible bond is the value of the security if it is
converted immediately. That is:
Conversion value = market price of common stock × conversion ratio = $40 × 22 = $880
b.
Market conversion price is the price that an investor effectively pays for the
common stock if the convertible bond is purchased:
Market conversion price = market price of the convertible bond/conversion ratio
= $1,050/22 = $47.73
14-13
24.
a.
i. The current market conversion price is computed as follows:
Market conversion price = market price of the convertible bond/conversion ratio
= $980/25 = $39.20
ii. The expected one-year return for the Ytel convertible bond is:
Expected return = [(end of year price + coupon)/current price] – 1
= [($1,125 + $40)/$980] – 1 = 0.1888 = 18.88%
iii. The expected one-year return for the Ytel common equity is:
Expected return = [(end of year price + dividend)/current price] – 1
= ($45/$35) – 1 = 0.2857 = 28.57%
b.
The two components of a convertible bond’s value are:

the straight bond value, which is the convertible bond’s value as a bond, and;

the option value, which is the value associated with the potential
conversion into equity.
i. In response to the increase in Ytel’s common equity price, the straight bond
value should stay the same and the option value should increase.
The increase in equity price does not affect the straight bond value component
of the Ytel convertible. The increase in equity price increases the option value
component significantly, because the call option becomes deep “in the money”
when the $51 per share equity price is compared to the convertible’s
conversion price of: $1,000/25 = $40 per share.
ii. In response to the increase in interest rates, the straight bond value should
decrease and the option value should increase.
The increase in interest rates decreases the straight bond value component
(bond values decline as interest rates increase) of the convertible bond and
increases the value of the equity call option component (call option values
increase as interest rates increase). This increase may be small or even
unnoticeable when compared to the change in the option value resulting from
the increase in the equity price.
14-14
25.
This strategy is a bear spread. The initial proceeds are: ¥900 – ¥300 = ¥600
The payoff is either negative or zero:
ST < 5000
5000 < ST < 6000
ST > 6000
Long call (X = 6000)
Short call (X = 5000)
0
0
0
– (ST – 5000)
ST – 6000
– (ST – 5000)
Total
0
– (ST – 5000)
–1000
Position
Breakeven occurs when the payoff offsets the initial proceeds of ¥600, which
occurs at a stock price of ST = ¥5600.
600
6000
0
26.
5000
ST
-400
Profit
-1000
Payoff
Buy a share of stock, write a call with X = 50, write a call with X = 60, and buy a
call with X = 110.
Position
Buy stock
Short call (X = 50)
Short call (X = 60)
Long call (X = 110)
Total
ST < 50
50 < ST < 60
60 < ST < 110
ST > 110
ST
ST
– (ST – 50)
ST
– (ST – 50)
– (ST – 60)
ST
– (ST – 50)
– (ST – 60)
ST – 110
0
0
0
ST
0
0
50
0
110 – ST
0
The investor is making a volatility bet. Profits will be highest when volatility is low
so that the stock price ends up in the interval between $50 and $60.
14-15